126 XIII INTEGRATION
Since the sequence (gn) is increasing and \ gn d[i^ I * gn dpi < + oo, the result
follows from (13.8.1).
(13.8.3) Let (fn) be a sequence of integr able functions, and suppose that there
/*
hdjn< 4-00 and — h ^fn :g h almost
every where, for alln. Then the functions lim inf/, andlim sup/, are integrable,
«-+ oo n~*oo
and we have
(13.8.3.1) Aiminf/B) d\i £ lim inf /„ d/j,
J \ n~*oo / n-*oo J
^ lim sup Ifndfj,^ (lim sup/J d\i.
n~*oo J J \ n-*-oo /
It is enough to prove that lim inffn is integrable and that the left-hand
n-»-oo
inequality in (13.8.3.1) is valid; the other assertions then follow by replacing
fn by — /, . Let gn = inf/n+ p . It follows from (1 3.8.2) applied to the sequence
p^O
(—fn+p)P£o and from ^e hypothesis fn+p ^ — h, that gn is integrable, and
clearly f gn d/j, £ (fn+p d\i for all p ^ 0, so that f gn d^ g inf (fn+p d\i. The
J J J p£QJ
sequence (gn) is increasing ; we have gn^h for all n, and by definition (1 2.7.1 0),
lim infyj, = sup gn . Hence it follows from (13.8.1) that lim inf/w is integrable
n-»oo n n-*oo
and that
(lim inf/J d/j, = sup gn dp ^ sup ( inf fn+p d/A
J \ n~*oo / n J n \p£0 J /
= lim inf /„ dfi.
w-»-oo J
(13.8.4) (Dominated convergence theorem) Let (fn) be a sequence of
integrable functions ; suppose that Iimfn(x) =f(x) exists almost everywhere
n-* oo
and that there exists a function g ^ 0 such that j g d/j, < +00 and \fn\ ^ g
almost everywhere, for alln. Then f is integrable and
(13.8.4.1) f/(x) dftx) .= lim (/„(*) d/i(x).
J B-+00-J
This follows from (13.8.3). In this situation the two extreme terms of
(13.8.3.1) are equal, whence the result.f:X-+Htobe integrable it is necessary and